Real conjugacy class: Difference between revisions

Revision as of 16:40, 9 May 2011

The term real conjugacy class is somewhat ambiguous and could refer to either of two things:

A conjugacy class of real elements, i.e., a conjugacy class with the property that every element in it is conjugate to its inverse.
The union of a conjugacy class and its inverse, i.e., an equivalence class under the relation of being real-conjugate.

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-==Definition==
+The term '''real conjugacy class''' is somewhat ambiguous and could refer to either of two things:
-A '''real conjugacy class''' in a [[group]] is defined as the union of any [[conjugacy class]] of the group with the conjugacy class of its inverse elements.
+* A [[conjugacy class]] of [[real element]]s, i.e., a conjugacy class with the property that every element in it is conjugate to its inverse.
+* The union of a conjugacy class and its inverse, i.e., an equivalence class under the relation of being ''real-conjugate''.
-===Alternative use===
-Note that the term ''real conjugacy class'' is sometimes also used for a conjugacy class of [[real element]]s, which means a conjugacy class in which every element is conjugate to its inverse. This is a ''different'' meaning and a more restrictive one.
-==Facts==
-===Relation with irreducible representations===
-* [[Number of irreducible representations over reals equals number of real conjugacy classes]]